How much gun violence was there really in the old west (US, late 1800s)

Almost none, outside of the very few sporatic instances of lawmen v outlaws, and most of these were not as hollywood shows them. Guns were everywhere in the west - but not on a cowboy's hip. Many towns had laws against openly carrying guns in public, however many people still kept pocket pistols [mostly cap and ball revolvers until the 1880s, and break top revolvers thereafter] in there coats.

Gun violence as depicted in hollywood normally has a good guy, a bad guy, a duel, and the understanding that he who draws first clears the other guy ina case of self defence. That's historical bull poop. First, there were no good guys in the few gun fights that happened. For example, Wyatt Earp was a violent pimp/ horse theif, Will Bill Hickok was a gambling addict/roughneck, and the Robin Hood figure of Jesse James was a cold blooded murdering pyscho who didn't give anyone anything. Gun fighters tended to be mentally ill lawmen or outlaws, who had stories built around them by dime novels. These dime novels took there short lives and made mountains out of mole hills.

Gun fights were not duels, they were violent murders, with the goal of putting as many bullets through the other person as possible, so that they would bleed to death quicker. Most happened with guns already out, normally both parties were drunk, and sometimes it took days for one or both to die. Gunfights were the on-foot drive bys of today. Wild Bill Hickok shot Dave Tutt in self defence, but he wasn't a cop at the time, and still was almost put behind bars. Not so romantic in real life.

Gun violence was rare, partly because towns tended to arrest or evict the homeless, and the unemployed [who are more likely to commit crimes, which is an unfair reality of life], people could pull up stakes and move away from their problems easier than today, where there are no safety valves for society in general. Less things were illegal in general, and the gang crimes and mass shooting we see today never had a rooting ground in the old west.